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Variational h-adaptation for strongly coupled problems
in thermo-mechanics
Rohit Pethe
To cite this version:
Rohit Pethe. Variational h-adaptation for strongly coupled problems in thermo-mechanics. Mechanical engineering [physics.class-ph]. École centrale de Nantes, 2017. English. �NNT : 2017ECDN0046�. �tel-02981707�
Rohit PETHE
Mémoire présenté en vue de lʼobtention
du grade de Docteur de lʼÉcole Centrale de Nantes
sous le sceau de l’Université Bretagne Loire
École doctorale : Sciences Pour l’Ingénieur, Géosciences, Architecture (SPIGA)
Discipline : Génie mécanique
Unité de recherche : Institut de Recherche en Génie civil et Mécanique
Soutenue le 14 Décembre 2017
Variational h-adaptation for strongly coupled
problems in thermo-mechanics
JURY
Président de jury : PONTHOT Jean-Philippe, Professeur, University of Liège
Rapporteurs : MOSLER Jörn, Professeur, TU Dortmund
FOURMENT Lionel, Professeur, Mines Paris Tech
Examinateurs : DRAPIER Sylvain, Professeur, Mines Saint-Étienne
RASSINEUX Alain, Professeur, Université de technologie de Compiègne
RACINEUX Guillaume, Professeur, Centrale Nantes
Directeur de thèse : STAINIER Laurent, Professeur, Centrale Nantes
´
Ecole Centrale de Nantes
Universit´
e bretagne loire
´
Ecole doctorale
Science Pour l’ing´
enieur, G´
eosciences, Architecture
Th`
ese de Doctorat
Sp´ecialit´e : M´ecanique des solides, des mat´eriaux, des structures et des
surfaces
Pr´esent´ee et soutenue par:
Rohit PETHE
Le – Novembre 2017 ´a l’´Ecole Centrale de Nantes
Variational h-adaptation for strongly coupled
problems in thermo-mechanics.
Jury
Rapporteurs J¨orn Mosler Technische universit¨at dortmund
Lionel Fourment Mines Paris Tech
Examinateurs Sylvain Drapier Mines Saint-´Etienne
Jean-Philippe Ponthot University of Li`ege
Alain Rassineux Universit´e de Technologie de Compi`egne
Guillaume Racineux Ecole Centrale de Nantes´
Directeur de th`ese Laurent Stainier
Co-encadrant Thomas Heuz´e
Acknowledgements
It has been a privilege to be at Centrale Nantes, first as a masters student and now as a PhD candidate.
I am grateful to my director of thesis Prof. Laurent Stainier who guided me all along my research with a lot of enthusiasm, without which this work wouldn’t have been possible. Apart from core research, I also learnt moral values because of his outlook towards life. I also thank my co-director of
thesis Dr. Thomas Heuz´e for his guidance and taking immense efforts to
polish my manuscript and research articles. I am indeed very much happy and proud to have an opportunity to work under Prof. Laurent Stainier and
Dr. Thomas Heuz´e.
I would also like to thank various faculty members like Prof. Nicolas M¨oes,
Prof. Anthony Nouy, Dr. Gr´egory Legrain, Mr. Alexis Salzmann whose
courses helped me build strong foundation in computational mechanics. I thank my former colleague Dr. Prashant Rai for thought provoking discussions about life and motivations. I must thank my colleagues Adrien Renaud, Jorge De Anda Salazar, Tauno Tiirats, Abdullah Waseem, Benoit Le, Romain Hamonou, Simon Paroissien, Alina Krasnobrizha, Daria Serbichenko, Quentin Auoul-Guilmard and various others who directly or indirectly helped me in completing the thesis.
I would like to remember the company of friends like Akash Kumar, Gaganpreet Kaur, Amey Rangnekar, Pranav Pandit, and others who would make my weekends fun and re-energize me for the next week.
I am thankful to my sister Rajashree Pethe who would happily clear my
doubts in C++. I should also thank my fianc´ee Avanti Shinde and her parents,
thank my friend Prathamesh Paigude from whom I learnt to be positive in any difficult situation. I also thank him for immense love and support towards me and my family.
Finally and most importantly I would like to thank my mother Manik Pethe, and my father Sanjay Pethe, for their blessings and active support all through my academic career and life so far, without whose sacrifices and initiatives, it would have been extremely hard to avail such a great opportunity of learning and experience.
Abstract
A mesh adaption approach for strongly coupled problems is proposed, based on a variational principle. The adaption technique relies on error indicated by an energy-like potential and is hence free from error estimates. According to the saddle point nature of this variational principle, a staggered solution approach appears more natural and leads to separate mesh adaption for mechanical and thermal fields. Using different meshes for different phenomena, precise solutions for various fields under consideration are obtained. Internal variables are considered constant over Voronoi cells, so no complex remapping procedures are necessary to transfer internal variables. Since the algorithm is based on a set of tolerance parameters, parametric analyses and a study of their respective influence on the mesh adaption is carried out. This detailed analysis is performed on uni-dimensional problems. The proposed method is shown to be cost effective than uniform meshing, some applications of the proposed approach to various 2D examples including shear bands and friction welding are presented.
Contents
1 Motivation and general context 15
1.1 Background . . . 15
1.2 Literature review . . . 18
1.2.1 Mesh adaptation . . . 18
1.2.2 Adaptation techniques . . . 19
1.2.3 Adaptation criteria . . . 20
1.2.4 Mesh adaptation for strongly coupled problems . . . . 21
1.2.5 Variational framework . . . 22
1.3 Objectives and challenges . . . 23
1.4 Conclusion . . . 25
2 Strongly coupled problems 28 2.1 Introduction . . . 28 2.2 Problems in multiphysics . . . 28 2.2.1 Strong coupling . . . 29 2.2.2 Weak coupling . . . 30 2.3 Solution schemes . . . 30 2.3.1 Monolithic approach . . . 30 2.3.2 Staggered approach . . . 31 2.4 Introduction to thermo-mechanics . . . 32 2.4.1 Balance equations . . . 32 2.4.2 Thermo-elasticity example . . . 34 2.4.2.1 Monolithic approach . . . 36
2.4.2.2 Staggered approach with isothermal split . . . 37
2.5 Conclusion . . . 38
3 Variational formulation in coupled thermo mechanics 42 3.1 Introduction . . . 42
3.2 Continuum modelling framework . . . 43
3.2.1 Free energy and dissipation potential . . . 43
3.2.2 Local evolution problem . . . 45
3.2.3 Variational formulation of the initial boundary value problem . . . 47
3.3 Time discrete modelling framework . . . 48
3.3.1 Local constitutive problem . . . 48
3.3.2 Incremental boundary value problem . . . 50
3.3.3 Dynamics . . . 51
3.4 Examples of constitutive models . . . 52
3.4.1 Purely thermal transient problem . . . 52
3.4.2 Linear thermo-elasticity . . . 52
3.4.3 Thermo-elasto-visco-plasticity . . . 53
3.5 Staggered algorithms . . . 54
3.6 Conclusion . . . 55
4 Mesh adaptation Algorithm 59 4.1 Introduction . . . 59
4.2 Local adaptation techniques . . . 60
4.2.1 Single Edge Bisection technique (SEB) . . . 60
4.2.2 A Backward Longest Edge Propagation Path (LEPP) Algorithm . . . 62
4.3 Adaption criteria . . . 65
4.3.1 Refinement . . . 65
4.3.2 Coarsening . . . 68
4.4 Equivalence with error norms . . . 70
4.5 Management of fields and internal variables . . . 74
4.5.1 Management of internal variables during adaption pro-cedure . . . 74
4.5.2 Interpolation of fields from one mesh to other . . . 75
4.6 Conclusion . . . 76
5 Unidimensional test cases 79 5.1 Introduction . . . 79 5.2 Steady state . . . 80 5.2.1 Analytical solution . . . 80 5.2.2 Numerical solution . . . 81 5.2.3 Cost analysis . . . 82 5.2.4 Parametric analysis . . . 83 5.2.5 Improved algorithm . . . 85 5.3 Thermo-elasticity . . . 87
5.3.1 Numerical solution fields . . . 87
5.3.2 Cost analysis . . . 89
5.4 Thermo-elasto-plasticity . . . 91
5.4.1 Numerical solution fields . . . 91
5.4.2 Analysis . . . 95
5.5 Conclusion . . . 96
6 Bidimensional test cases 99 6.1 Introduction . . . 99
6.2 Steady state thermal . . . 100
6.3 Transient purely thermal test case . . . 102
6.4 Linear thermo-elasticity . . . 104
6.5 Shear bands . . . 107
6.6 Linear Friction Welding (LFW) . . . 112
6.7 Conclusion . . . 118
List of Figures
1.1 High speed impact of a metal bar [1]. . . 16
1.2 Finite element solution with fixed mesh produces highly
dis-torted elements [1]. . . 16
1.3 Channel flow simulation using Eulerian approach [1]. . . 17
1.4 ALE simulation of metal bar impact problem [1]. . . 17
4.1 The edge identified is BD. Therefore, the patch contains two
elements adjacent to edge BD. The refined version of patch
contains the new node E. . . 61
4.2 The edge identified is AB, which is on the boundary. Therefore,
the patch contains a single element DAB. The refined version
of the patch contains the new node E. . . 61
4.3 The edge identified is BD. Therefore, the patch contains two
elements adjacent to edge BD. The refined version of patch
contains four new nodes shown in green. . . 61
4.4 The edge identified is AB which is on the boundary. Therefore,
the patch contains the single element DAB. The refined version
of patch contains three new nodes shown in green. . . 61
4.5 Original algorithm of Rivara [61] based on target triangle.
Target triangle is t0. . . 63
4.6 The proposed algorithm based on target edge. Target edge is
ED. . . 64
4.7 The LEPP algorithm gives the single step refinement of several
edges. They are coarsened one by one. Here edge AGC is
4.8 The diagram on the left shows a triangular parent element with three integration points shown in different colours. The pieces of Vorono¨ı cells intersected with triangle corresponding to each Gauss point are shown in the figure in the middle with respective colours. These Vorono¨ı cells represent the domain of influence of each Gauss point. The diagram on the right shows the bisected triangle. Two new triangles are formed as a result of bisection. The children Gauss points inherit data from parent Gauss points shown in same colors in the figure
above. . . 75
4.9 Figure on the left shows an element ABC and one of its Gauss points on which fields from the other mesh are to be interpo-lated. The figure in the middle shows the other mesh and the element ABC in dotted blue. The element in which Gauss point of element ABC lies is identified as element DEF as shown in the figure on the right. Therefore, the external fields can be interpolated from nodal values at nodes D, E and F . Variables at integration points are inherited from the closest integration point in element DEF which is shown in red. . . . 76
5.1 Analytical solution. . . 81
5.2 Numerical solution on the initial mesh. . . 81
5.3 Numerical solution on an intermediate mesh. . . 82
5.4 Numerical solution on the final mesh. . . 82
5.5 L2 norm error in temperature field with respect to number of nodes. . . 83
5.6 Energy norm error in energy like potential with respect to number of nodes. . . 83
5.9 Effect of T olr when T ol0 is 10−2 and T old is 10−4. . . 84
5.10 Effect of T olr when T ol0 and T old are fixed to 10−2. . . 84
5.7 Effect of T ol0 when T olr and T old are fixed to 10−4. . . 84
5.8 Effect of T ol0 when T olr and T old are fixed to 0.5. . . 84
5.12 Effect of T old when T ol0 is 10−2 and T olr is 10−2. . . 86
5.13 Comparison of L2 error in Temperature analysis between origi-nal and improved algorithm. . . 86
5.14 Comparison of energy error analysis between original and im-proved algorithm. . . 86
5.15 Parametric analysis of T olu represented in terms of L2 error in Temperature. . . 87
5.16 Parametric analysis of T olu represented in terms of energy error. 87 5.17 Displacement field at time = 1 second. . . 88
5.18 Temperature field at time = 1 second. . . 88
5.19 Displacement field at time = 50 seconds. . . 88
5.20 Temperature field at time = 50 seconds. . . 88
5.21 Displacement field at time = 301 seconds. . . 89
5.22 Temperature field at time = 301 seconds. . . 89
5.23 Cost analysis of mechanical mesh at time=1 second. . . 90
5.24 Cost analysis of thermal mesh at time=1 second. . . 90
5.25 Cost analysis of mechanical mesh at time=50 seconds. . . 90
5.26 Cost analysis of thermal mesh at time=50 seconds. . . 90
5.27 Cost analysis of mechanical mesh at time=301 seconds. . . 91
5.28 Cost analysis of thermal mesh at time=301 seconds. . . 91
5.29 Geometry of the test case [31]. . . 92
5.30 Reference solution in displacement [31]. . . 93
5.31 Reference equivalent plastic strain [31]. . . 93
5.32 Reference solution in temperature. . . 94
5.33 Equivalent plastic strain analytical[31] and numerical. . . 94
5.34 Solution on adapted mesh at rotation θ = 4◦ of outer cylinder. 94 5.35 Solution on adapted mesh at rotation θ = 5◦ of outer cylinder. 94 5.36 Solution on adapted mesh at rotation θ = 6◦ of outer cylinder. 94 5.37 Solution on adapted mesh at rotation θ = 9◦ of outer cylinder. 94 5.38 L2 error analysis at rotation θ = 4◦ of outer cylinder. . . 95
5.39 L2 error analysis at rotation θ = 5◦ of outer cylinder. . . 95
5.40 L2 error analysis at rotation θ = 6◦ of outer cylinder. . . 96
6.1 Solution field on initial mesh along with the boundary conditions.101
6.2 Numerical solution on adapted mesh. . . 101
6.3 Analysis of the algorithm. Number of nodes on X axis and L2 error in temperature on Y axis. . . 101
6.4 Representation of heating area at an instant. . . 102
6.5 Temperature field at time step 1. . . 103
6.6 Temperature field at time step 8. . . 103
6.7 Temperature field at time step 24. . . 103
6.8 Temperature field at time step 40. . . 103
6.9 Temperature field at time step 48. . . 103
6.10 Temperature field at time step 60. . . 103
6.11 Geometry and boundary conditions for thermo-elasticity prob-lem. σnt = 0 on all boundaries. . . 104
6.12 Initial thermal and mechanical mesh for thermo-elasticity prob-lem. . . 104
6.13 Stress magnitude along the radial direction on the x-ligament. 105 6.14 Magnitude of gradient of temperature along the radial direction on the x-ligament. . . 105
6.15 Mechanical adapted mesh obtained using single edge bisection technique. . . 106
6.16 Mechanical adapted mesh obtained using Rivara’s technique. . 106
6.17 Thermal adapted mesh obtained using single edge bisection technique. . . 106
6.18 Thermal adapted mesh obtained using Rivara’s technique. . . 106
6.19 Comparison of mesh adaption techniques for mechanical mesh. 107 6.20 Comparison of mesh adaption techniques for thermal mesh. . . 107
6.21 Geometry and mechanical boundary conditions of shear band specimen. . . 108
6.22 Initial mesh for thermal and mechanical parts. . . 108
6.23 Final adapted thermal mesh with temperature field. . . 109
6.24 Final adapted mechanical mesh with equivalent plastic strain. 109 6.25 Evolution in time of length of segment l for k = 0.3 in equation (6.3). . . 110
6.26 Evolution of hσntiAB with time. . . 111
6.27 Evolution of maximum temperature Tmax with time. . . 111
6.28 Numerical and analytical temperature profile along n at time 0.103037. . . 112 6.29 Numerical and analytical rate of plastic strain along n at time
0.103037. . . 112 6.30 Geometry and boundary conditions of the linear friction
weld-ing problem in [41]. . . 114 6.31 Simplified modeling of the problem. Boundary conditions of
symmetry on AD, contact and heat flux on AB and imposed displacement on DC are applied. Boundary BC is free. . . 114 6.32 Initial mesh for thermal part for linear friction welding test case.114 6.33 Initial mesh for mechanical part for linear friction welding test
case. . . 114 6.34 Stress field and mechanical mesh (along with reflection) at
preliminary stages. . . 115 6.35 Mechanical mesh (along with reflection) after developed stress
field. . . 115
6.36 Stress field at last calculated time step along. . . 115
6.37 Temperature field and thermal mesh (along with reflection) at preliminary stages. . . 116 6.38 Thermal mesh (along with reflection) at final time step. . . 116 6.39 Temperature profile on a line along y direction passing through
mid-points of segments AB and DC in figure 6.31. . . 116
6.40 Magnitude of stress tensor on a line along y direction passing through mid-points of segments AB and DC in figure 6.31. . . 116 6.41 If the initial patch is ABCD as shown in the diagram on the
left. The refined patch on which local problem will be solved is shown in the middle diagram. If the error indicated is highly significant, even more refined version of patch would be used in the global mesh as shown in the diagram on the right. . . . 123
6.42 Diagram on left shows an original triangle which can be divided into three by inserting a new node at its centroid as shown in diagram on right. . . 124
Introduction g´
en´
erale
La concurrence sur le march´e a oblig´e les entreprises `a acc´el´erer le processus
de conception, de prototypage, de fabrication et de lancement de nouveaux
produits sans compromettre leur qualit´e. Pour atteindre cet objectif, elles
doivent repousser les limites technologiques. L’´emergence d’´equipements de
prototypage rapide tels que les imprimantes 3D r´esulte de ces efforts.
Toute-fois, tous les produits ne peuvent pas ˆetre prototyp´es rapidement ni rentables.
Une technologie qui peut permettre d’´eliminer ou plutˆot de r´eduire le besoin de
prototype est donc n´ecessaire. La croissance des technologies de l’information
et de l’informatique a augment´e le pouvoir de calcul comme jamais auparavant.
En cons´equence, un nouveau domaine de la science informatique a ´emerg´e
en compl´ement des sciences th´eoriques et exp´erimentales traditionnelles. La
m´ecanique num´erique est une science informatique qui traite des solutions
ap-proch´ees aux probl`emes de m´ecanique en utilisant des techniques num´eriques.
Ainsi, les simulations num´eriques permettent de r´eduire le besoin de
proto-types, en fournissant des donn´ees virtuelles, simul´ees du comportement de
tout ou partie d’un produit. Dans un monde id´eal, une simulation num´erique
devrait produire des r´esultats tr`es proches du monde r´eel. Dans le monde r´eel,
chaque situation implique plusieurs ph´enom`enes physiques. Par exemple, dans
un moteur `a combustion, l’´energie chimique du carburant est convertie en
chaleur, qui est ensuite convertie en pression fluide qui, `a son tour, produit le
mouvement m´ecanique. Le processus implique l’effet de ph´enom`enes
chim-iques, thermchim-iques, fluides et structurels les uns sur les autres. Une m´ethode
num´erique qui incorpore ces int´eractions sera capable de produire des r´esultats
peut out repr´esenter, et est g´en´eralement produite en regard d’un objectif
pr´ecis, et donne des r´esultats valides dans un domaine born´e.
Une m´ethode num´erique tr`es populaire pour les probl`emes structurels est
la m´ethode des ´el´ements finis. La m´ethode est bas´ee sur une approche
lagrang-ienne o`u un maillage repr´esentant la g´eom´etrie suit les points mat´eriels de la
structure consid´er´ee. Pour les probl`emes impliquant de grandes d´eformations,
une distorsion de maille s´ev`ere est observ´ee, pouvant conduire un jacobien
n´egatif, et donc une solution non physique.
Par cons´equent, une approche envisag´ee depuis dj fort longtemps est
d’adapter la forme et la taille des mailles au cours de la transformation. Cela
permet d’´eviter le probl`eme des distorsions s´ev`eres des mailles. De plus,
en simulant des processus industriels fortement coupl´es comme le forgeage,
l’usinage, le soudage par friction, etc., les effets dynamiques et transitoires
font que les domaines d’int´erˆet changent rapidement. Dans le cadre de la
m´ethode des ´el´ements finis, cela signifie que les domaines d’int´erˆet changent
de position au cours du temps. Par cons´equent, une strat´egie d’adaptation de
maillage permet d’obtenir des solutions pr´ecises `a chaque instant en affinant le
maillage dans les domaines d’int´erˆet et en d´eraffinant le maillage dans d’autres
domaines. Dans ce travail, nous pr´esentons une strat´egie d’adaptation de
maillage bas´ee sur un principe variationnel d´edi´es aux probl`emes fortement
coupl´es en thermo-m´ecanique.
Les strat´egies d’adaptation sont bas´ees sur des techniques et des crit`eres
d’adaptation. Les techniques d’adaptation traitent essentiellement des
as-pects g´eom´etriques de l’adaptation. Les crit`eres d’adaptation refl`etent la
particularit´e du probl`eme `a l’´etude. Les proc´edures d’adaptation de maillage
globale cr´eent un maillage compl`etement nouveau et utilisent des proc´edures
de projection pour transf´erer des variables internes [50, 58]. Les m´ethodes
bas´ees sur le remaniement global du domaine d’int´erˆet n´ecessitent le transfert
de variables internes entre les mailles, ce qui peut conduire `a une diffusion
artificielle de ce dernier. Rivara et al. [62, 61, 60] propose des mises `a jour
explicites pour les changements de maillage locaux. Grinspun et al. [26]
ont propos la m´ethode CHARMS pour le raffinement de maillage de facon
bas´es sur des estimations d’erreur ou l’asym´etrie du maillage. L’estimation
d’erreur de Z2 couramment utilis´ee depuis Zienkiewicz et Zhu [74] utilise les
contraintes dans un ´el´ement et est bas´ee sur un processus de reconstruction
de champ pour obtenir un champ de contrainte de r´ef´erence. La diff´erence
dans l’´el´ement entre les champs fournit une estimation d’erreur bas´ee sur
le d´egrad´e. La majorit´e des crit`eres d’adaptation des mailles propos´es dans
la litt´erature sont bas´es sur l’estimation d’erreurs. Dans ces m´ethodes, la
strat´egie consiste `a adapter le maillage pour minimiser une erreur li´ee; ou
par une application r´ecursive des ´etapes de raffinement local [71, 2]. Mais ces
m´ethodes ont certaines limites: ils fonctionnent bien avec des mod`eles
constitu-tifs lin´eaires (par exemple l’´elasticit´e), mais deviennent plus complexes lorsque
des mod`eles constitutifs non lin´eaires sont utilis´es. En outre, les champs
admissibles doivent ˆetre reconstruits [37, 74]. De plus, les bornes d’erreur
standard n´ecessitent une certaine r´egularit´e de la solution pour assoir leur
validit´e [17]. Par cons´equent, il peut ˆetre difficile et coˆuteux d’utiliser cette
approche pour des probl`emes complexes impliquant des mod`eles constitutifs
non lin´eaires et / ou de grandes d´eformations. Une proc´edure d’adaptation
de maillage alternative pour des probl`emes purement m´ecaniques a ´et´e
pro-pos´ee par Mosler et al. [48, 49], bas´ee sur une approche variationnelle de
[51]. Cette technique utilise un indicateur d’erreur plutˆot qu’un estimateur
d’erreur. Dans une approche variationnelle, un potentiel d’´energie doit ˆetre
minimis´e (ou maximis´e), dont la valeur scalaire indique un certain niveau de
qualit´e de l’approximation effectu´ee. Aucune estimation d’erreur n’est utilis´ee
`a n’importe quel stade de l’algorithme. Cette approche permet l’adaptation
de maillages en pr´esence de grandes d´eformations et de comportements non
lin´eaires.
Alors que l’adaptation des mailles `a l’aide d’estimateurs d’erreur est
bien ´etablie pour les probl`emes impliquant une seule physique, seules quelques
tentatives ont ´et´e propos´ees en ce qui concerne les m´ethodes d’adaptation
de maillage pour des probl`emes fortement coupl´es. La plupart des m´ethodes
disponibles dans la litt´erature adaptent le maillage pour un seul des champs
Dans le pr´esent travail, nous pr´esentons une strat´egie d’adaptation de
maillage pour des probl`emes fortement coupl´es bas´es sur une approche
vari-ationnelle, et plus particuli`erement des probl`emes thermo-m´ecaniques. En
effet, les principes variationnels sont bas´es sur la minimisation ou la
maximi-sation d’une fonctionnelle, dont la valeur locale s’av`ere ˆetre un bon indicateur
d’erreur num´erique sur un patch d’´el´ements finis. Les objectifs du pr´esent
travail peuvent ˆetre ´enum´er´es comme suit:
1. Proposer un algorithme de h−adaptation pour des probl`emes fortement
coupl´es en thermo-m´ecanique. Le d´efi dans les probl`emes fortement
coupl´es est que les deux ph´enom`enes, m´ecaniques et thermiques
peu-vent ˆetre `a des ´echelles spatio-temporelles tr`es diff´erentes. En outre, les
emplacements spatiaux des domaines d’int´erˆet pour les deux champs
peu-vent ˆetre tr`es diff´erents. L’approche d’adaptation de maillage propos´ee
repose sur une approche d´ecal´ee [3] coupl´ee avec des maillages diff´erents
pour diff´erents champs. L’adaptation s´equentielle de diff´erents
mail-lages permet de capturer les diff´erentes ´echelles spatiales des diff´erents
champs.
2. L’approche propos´ee par Mosler et al. [48, 49] pour des probl`emes
uniphysique permet de s’affranchir d’estimateurs d’erreur, et donc de
reconstructions coˆuteuse de champs. L’approche propos´ee a pour objet
d’utiliser cet avantage et de l’´etendre `a des probl`emes coupl´es. Par
cons´equent, le probl`eme repose sur un indicateur d’erreur bas´e sur la
valeur de la fonctionnelle variationnelle.
3. ´Eviter la diffusion num´erique excessive caus´ee par les proc´edures de
projection complexes pour transf´erer des variables internes d’un maillage
`
a l’autre. En effet, les m´ethodes bas´ees sur le remaniement global du
domaine d’int´erˆet provoquent une diffusion num´erique significative, elles
ne sont donc pas consid´er´ees dans ce travail. Plutˆot, les techniques
locales d’adaptation des mailles bas´ees sur la bisection du bord sont
pr´ef´er´ees. Dans le pr´esent travail, les techniques d’adaptation ne sont
r´epartition constante des variables internes sur les cellules Vorono¨ı, des
proc´edures de projection complexes provoquant une diffusion num´erique
significative peuvent ˆetre ´evit´ees. Cet op´erateur de transfert a ´et´e ´etudi´e
en d´etail par Ortiz et Quigley [50].
4. La m´ethode devrait ˆetre facilement adaptable pour diff´erents probl`emes.
Diff´erents cas test d’abord unidimensionnels puis bidimensionnels sont
pr´esent´es aux chapitres 5 et 6, notamment des probl`emes thermiques
sim-ples pour une illustration simple de la m´ethode, des probl`emes fortement
coupl´es associ´es `a la thermo-´elasticit´e, un probl`eme de simulation du
ph´enom`ene de bande de cisaillement et un cas repr´esentatif de soudage
par friction. Ces applications ´etendues d´emontrent l’adaptabilit´e de
l’algorithme `a diff´erents probl`emes.
5. L’algorithme propos´e doir ˆetre efficient pour des probl`emes suffisamment
complexes, c’est-`a-dire moins coteux `a pr´ecision donn´ee qu’un raffinage
uniforme, pour ˆetre utile dans des applications pratiques. Cette analyse
des cots est effectu´ee sur diff´erents cas test dans les chapitres 5 et 6 en
profondeur et il est ´etabli que l’algorithme est efficient par rapport `a un
Chapter 1
Motivation and general context
1.1
Background
Innovation has been the key for successful product development for centuries.
From the second part of 20th century, competition in the market has forced
companies to accelerate the process of designing, prototyping, manufacturing and launching of new products without compromising the product quality. Achieving this target requires to push technological boundaries. Emergence of rapid prototyping equipments such as 3D printers is an example of outcome of these efforts. However, neither every product can be rapid prototyped nor it is cost-effective. A technology that can either eliminate or reduce the need to prototype is thus needed. The growth in information technology and computer science has increased computational power like never before. As a result, a new field of computational science has emerged as a new leg of science, in addition to theoretical and experimental ones. Computational mechanics is a computational science that deals with finding approximate solutions to problems in mechanics by using numerical techniques. Thus, numerical simulations fulfill the need of technology that can reduce the need of prototypes. A numerical simulation is designed to produce results that are very close to the real world. In real world, every situation involves several physical phenomenon. For example in an internal combustion engine, the chemical energy of fuel is converted into heat, which is then converted to fluid pressure which in turn produces the mechanical movement. The process
involves effect of chemical, thermal, fluid and structural phenomena on each other. A numerical method that incorporates these interactions will be able to produce accurate results.
A very popular numerical method for structural problems is the finite element method. The method is based on a Lagrangian approach where a mesh is created that represents the geometry and the mesh follows material points. However, for problems involving large strains, severe mesh distortion may occur. For example, consider a simulation involving a high speed impact of metal bar as shown in figure 1.1. When simulated using a finite element approach with a mesh stuck on the body, a highly distorted mesh is obtained as shown in figure 1.2. This poses several numerical difficulties involving flat elements, Jacobian going to zero or negative etc.
Figure 1.1: High speed impact of a metal bar [1].
Figure 1.2: Finite element solution with fixed mesh produces highly dis-torted elements [1].
In order to overcome this difficulty, several authors have proposed different methods. First is an Eulerian method in which the mesh remains fixed in space. That is, material points flow through the mesh. This approach is suitable for problems in fluid mechanics such as flow problems. The channel flow problem depicted in figure 1.3 demonstrates this approach. In order to use this approach for structural problems, a very fine mesh would be needed to capture material response making the method computationally very expensive. In addition, transport of history variables causes loss in information
Figure 1.3: Channel flow simulation
using Eulerian approach [1]. Figure 1.4: ALE simulation of metal
bar impact problem [1].
and therefore accuracy. Second, are ALE (Arbitrary Lagrangian Eulerian) approaches in which the mesh inside the domain can move arbitrarily to optimize the shape of elements while the mesh on the boundaries and the interfaces of the domains can move along the materials to precisely track the boundaries and interfaces of a multi-material system. For metal bar impact problem, this is illustrated in figure 1.4. While it has the advantage that it allows smoothing of distorted mesh, the main difficulty is the path dependent behaviour of the plastic flow being modelled. Due to the path dependence, the relative motion between the mesh and the material must be accounted for in the material constitutive equations. In addition, the ALE method does not allow new (damaged) surfaces to be created and is limited to geometries where the material flow is relatively predictable [1]. Third, meshless methods are also proposed that (theoretically) do not depend on any mesh, but face difficulties in application of boundary conditions and robustness.
Therefore, one common idea is to consider mesh adaptation in the frame-work of finite element method. This allows to avoid the problem of severe mesh distortions. In addition, while simulating strongly coupled industrial process like forging, machining, friction welding, etc., dynamic and transient effects cause the domains of interest to change rapidly. In the framework of finite element method, this means that the domains of interest change their spatial location with time. Therefore, a mesh adaptation strategy also allows to obtain precise solutions at each time step by refining the mesh in
domains of interest and coarsening the mesh in other domains. In this work, we present a mesh adaptation strategy based on a variational principle for strongly coupled problems in thermo-mechanics.
The purpose of this chapter is to introduce the background of the problem, make a brief state of the art of mesh adaption, and define objectives and challenges. In order to put the work in context, a comprehensive literature review is presented which revisits the major work in mesh adaptation tech-niques, mesh adaptation criteria also the variational framework on which the proposed mesh adaptation technique is based. Then the objectives of the thesis along with the challenges are explained.
1.2
Literature review
1.2.1
Mesh adaptation
Numerical error in finite element approximation is related to the mesh size h, the degree of polynomial appearing in the element shape function p and regularity of the solution r as follows:
||e||Hm ≤ chα||u||Hr (1.1)
where Hk is the space of functions possessing k square integrable derivatives,
|| · ||Hk represents the Sobolev norm, c is a constant (element-dependent), u
represents the exact solution, and the exponent α is given by:
α = min(p + 1 − m, r − m) (1.2)
Mesh adaptation is a way to exploit this mathematical result in order to focus efforts in the domain of interest and release in other areas. Mesh adaptation processes can be divided into three broad categories. First is h-adaptation where the mesh size is optimized [21]. This means, element size is adapted using the same type of elements. Size is reduced where interpolation must be enriched to achieve better accuracy. On the other hand, size is increased where solution is sufficiently accurate. It may contain processes of element refinement and/or coarsening, thereby increasing and/or reducing the number
of degrees of freedom. Second, r-adaptation where the number of nodes are same but their location and connectivity is changed [39, 56]. The strategy simply involves redistribution of the nodes. This has advantage of preserving number of degrees of freedom thereby not increasing the computational cost. The data structure and the coding remains straightforward as it simply involves changes in node co-ordinates and connectivities. Third, p-adaptation where the order of interpolation polynomial within an element is changed [52]. The strategy involves using a fixed finite element mesh and adapting the interpolation order of the elements. In this strategy, the convergence
to the exact solution is dictated by chα if the exact solution is smooth as
seen from equation (1.1). A hybrid of h-adaptation and p-adaptation is also proposed in the literature, and called hp-adaptation [57]. In this strategy, along with mesh size, interpolation order is also adapted. The purpose of the present work is to introduce an approach based on h-adaptation, however, the extension of the strategy to p-adaptation is quite straightforward.
Mesh adaptation strategies depend on adaptive techniques and adaptation criteria. Adaptive techniques essentially deal with geometric aspects of adap-tation. On the other hand, the adaptation criteria captures the peculiarities of the problem under consideration. The following subsections make a brief review of adaptation techniques for h-adaptation.
1.2.2
Adaptation techniques
Several mesh adaptation techniques have been proposed in the literature. Rivara et al. [62, 61, 60] propose explicit updates for local mesh changes. Molinari et al. [47] use local coarsening and refinement method based on mesh size for shear bands. Mesh adaptation for shear bands has also been studied in plane strain [7, 11]. Global mesh adaptation procedures create a completely new mesh and use remapping procedures to transfer internal variables [50, 58]. Using gradient based indicators, global remeshing technique has been applied to impact problems [22]. Global remeshing techniques also handle mesh distortions in machining problems [45]. Methods based on global remeshing of the domain of interest require to transfer internal variables between meshes,
which can lead to artificial diffusion of the latter unless specific methods are used [13]. Camacho et al. [15] propose remeshing methods using advancing front methods for ballistic penetration problems. In [19, 69], authors use mesh adaptation for shape optimization of structures. Grinspun et al. [26] proposed CHARMS method for hierarchical mesh refinement.
1.2.3
Adaptation criteria
Classically, the mesh adaptation criteria have been based on error-estimates
or mesh skewness. The commonly used Z2 error estimate proposed by
Zienkiewicz and Zhu [74] uses stresses within an element and is based on a recovery process to obtain reference stress. The difference between element and reference stress provides gradient based error estimate. Curvature based error estimates have been proposed by Borouchaki et al. [12]. Error estimates based on constitutive relation error have also been studied [38, 18, 36]. In these methods, the finite element solution is described as a displacement-stress pair such that the displacements satisfy kinematic constraints like boundary conditions and initial conditions while the stresses satisfy the equilibrium conditions. The displacements and stresses do not satisfy the constitutive relations (stress-strain relations) which provides an error measure which they refer to as the constitutive relation error. Romero et al. [63] propose an error estimate based on time update. Gurtin [27] uses configurational forces for r-adaptation. Some authors also use gradients of physical quantities as mesh adaptation criteria [7, 11, 50]. Error estimates can also be based on variational principles [33, 34, 16, 58]. Many other error estimators are studied by various researchers [4, 35, 42, 24].
In the fluid mechanics community, the main emphasis is on the proper resolution of flow field. Therefore, in order to capture boundary layers, shock waves and high speed compressible flows, mesh adaptation techniques are necessary. Significant amount of work has been done for mesh adaptation for compressible flows [44, 54, 43, 29]. Most of these mesh adaptation criteria are based on error estimates from gradients of flow fields. In the calculation of lift
and drag of an airfoil in presence of shocks and viscous effects, error estimators based on bounds on functional outputs have been proposed [53, 70].
A majority of mesh adaptation criteria proposed in the literature are based on error-estimation. In these methods, the strategy is to adapt the mesh to minimize an error bound among all meshes of fixed size; or by recursive application of local refinement steps [71, 2]. But these methods have certain limitations. They work well with linear constitutive models (for example elasticity), but become more complex when non-linear constitutive models are used. Moreover, admissible fields need to be reconstructed [37, 74]. In addition, standard error bounds require a certain regularity of the solution for their validity [17]. Therefore, it can be difficult and costly to use this approach for complex problems involving non-linear constitutive models and/or large deformation.
An alternative mesh adaptation criterion for purely mechanical problems was proposed by Mosler et al. [48, 49], based on the variational approach of [51]. This technique uses an error indicator rather than an error estimator. In a variational approach, an energy like potential is to be minimized (or maximized), the scalar value of which indicates the level of approximation following the minimum (or maximum) criterion. No error estimates are used at any stage of the algorithm. It allows mesh adaptation in presence of large deformations and non-linear constitutive behavior. In addition, it was shown in [48] that variational h-adaptation could be combined with variational r-adaptation, at least for hyperelastic behavior. Indeed, r-adaptation would involve remapping in the presence of internal variables, and was not considered by these authors for dissipative behaviors. In [48, 49], the authors addressed isothermal, steady state mechanical problems.
1.2.4
Mesh adaptation for strongly coupled problems
While mesh adaptation using error estimation is well established for single field problems, only few attempts have been made towards mesh adaptation methods for strongly coupled problems. Most of the methods available in literature adapt the mesh for only one of the considered fields [5]. Solin et
al. [64] use multimesh adaptation approach for weakly coupled problems, but the method is limited to thermo-elasticity. Vokas et al. [72] consider a single mesh and h-refinement affects all fields simultaneously, therefore the method fails to capture different scales and spatial resolutions of different fields. Moreover, the mesh adaptation criteria relies on error estimators that work well with linear constitutive models, but are very complex in the case of non-linear constitutive models due to their need to reconstruct admissible fields. Therefore, it appears difficult and expensive to use this approach for strongly coupled problems with non-linear constitutive models and/or large deformation. Ramadan et al. [59] propose a bimesh method for strongly coupled thermo-mechanical problems with localized deformations. These authors propose a method in which a thermal mesh is uniformly fine; whereas, the mechanical mesh uses the thermal mesh in the deformation zone and coarsened version of thermal mesh in zone of insignificant deformation. This strategy was mainly proposed to speed up the calculations as the mechanical problem is quite expensive.
1.2.5
Variational framework
In the present work, we present a strategy of mesh adaptation for strongly cou-pled problems based on a variational approach. Indeed, variational principles are based on minimization or maximization of a functional, the local value of which turns out to be a good indicator of numerical error on a patch of finite elements. This idea has been proposed and exploited for purely structural problems by Mosler et al. [48, 49]. Its extension to strongly coupled problems requires an associated variational formulation. Thermo-elastic and thermo-visco-elastic problems have been extensively investigated in [10, 9, 6, 8, 30, 46]. But, formulations for coupled thermo-mechanical problems involving non-linear dissipative behaviour, such as thermo-elasto-visco-plasticity have been recently summarized by Stainier [66, 73], which itself is an extension of former work on variational visco-plastic constitutive updates proposed by Stainier and Ortiz [51].
1.3
Objectives and challenges
The major objective of the thesis is to propose an h-adaptation algorithm for strongly coupled problems in thermo-mechanics. The challenge in strongly coupled problems is that the two phenomena, mechanical and thermal can develop at very different spatial and temporal scales. Moreover, the spatial locations of domains of interest for the two fields can be very different. Therefore, it is very difficult to have a single adaptive mesh that can effectively capture both fields. The proposed mesh adaptation approach relies on a staggered approach [3] coupled with different meshes for different fields. Sequential adaptation of different meshes allows to capture different scales and spatial resolutions of different fields. The second chapter of the thesis gives an introduction to multi-physics and solution schemes in general setting. Then a specific problem of thermo-mechanics is taken into consideration where monolithic and staggered approaches are explained in detail.
The second objective is that the mesh adaptation criteria should be free from any error estimates because of the drawbacks mentioned in the previous section. In order to achieve this objective, the proposed approach extends the approach proposed by Mosler et al.[48, 49] for single field problems. Therefore, the problem relies on an error indicator based on the value of a variational functional. The proposed approach exploits this thermo-mechanical varia-tional formulation of [66, 73]. The third chapter presents this variavaria-tional framework for thermo-mechanical problems. These formulations allow to represent the thermo-mechanical problem as a saddle point problem. First, a continuum model is presented followed by a time discrete model. Some examples of constitutive models are also presented.
The third objective is to avoid the excessive numerical diffusion caused by the complex remapping procedures to transfer internal variables from one mesh to another. Because the methods based on global remeshing of the domain of interest cause significant numerical diffusion, they are not considered. Instead, local mesh adaptation techniques based on edge bisection are preferred. In the present work, adaptation techniques are only studied for triangular elements. Assuming a constant distribution of internal variables over Vorono¨ı cells,
complex remapping procedures causing significant numerical diffusion can be avoided. This transfer operator was studied in detail by Ortiz and Quigley [50] and is presented in Chapter 4. The same holds true between the steps of the staggered scheme, avoiding significant numerical diffusion assuming a constant distribution of internal variables over elementary cells consisting of the intersection of Vorono¨ı cells and triangular elements. Indeed, two mesh adaptation techniques are presented. First, a Single Edge Bisection (SEB) technique [48] is considered, allowing for anisotropic meshes. Second, Rivara’s Longest Edge Propagation Path (LEPP) technique [62] is used which constrains the element aspect ratio. One could also extend this strategy for techniques like CHARMS [26] but this extension was not exploited here. The fourth chapter presents the proposed mesh adaptation algorithm. The considered adaptation techniques are presented in details followed by mesh adaptation criteria that depend on the variational framework presented in the third chapter.
Fourth, the proposed algorithm should be cost effective in case of suffi-ciently complex problems with respect to using a single uniform mesh for it to be useful in practical applications. An extensive cost analysis is performed on different test cases and it is established that the algorithm is indeed cost effective with respect to using single uniform mesh in case of sufficiently com-plex problems even using the most pessimistic cost estimate for the adaptive algorithm. Chapter 5 presents unidimensional test cases where analysis of the proposed algorithm is presented, demonstrating the cost effectiveness of the algorithm for complex problems. An extensive parametric analysis of the algorithm parameters is also presented in chapter 5.
The fifth objective is that the method should be easily adaptable for different problems. Different bidimensional test cases are presented in Chapter 6 including simple thermal problems, strongly coupled problems in thermo-elasticity, a problem simulating shear band phenomenon and a representative case of friction welding. These wide applications demonstrate the adaptability of the algorithm to different problems.
1.4
Conclusion
In this chapter the mesh adaptation for strongly coupled problems was motivated. Following the literature review, the objectives and challenges of the thesis were stated. Strategies to achieve the proposed objectives were introduced and relevant bibliography was extracted.
Probl`
emes fortement coupl´
es
Un probl`eme qui implique de nombreux ph´enom`enes physiques s’influen¸cant
les uns les autres est appel´e un probl`eme multiphysique coupl´e. Consid´erons
par exemple une bande de cisaillement dans laquelle la dissipation d’´energie
m´ecanique provoque une ´el´evation de la temp´erature, laquelle am`ene le
mat´eriau `a ramollir, ce qui conduit `a des contraintes de cisaillement plus
faibles. Pour r´esoudre ce probl`eme, les ing´enieurs doivent combiner des
mod`eles physiques et des algorithmes, contenant tous les ph´enom`enes physiques
en pr´esence pour rendre compte de l’effet observ´e. Pour beaucoup de probl`emes
de l’ing´enieur, les effets physiques d’int´erˆet r´esultent de la combinaison de
plusieurs physiques. Il est souvent rappel´e que tous les probl`emes devraient
ˆ
etre suppos´es coupl´es avant (pour certains) pour infirmer cette proposition.
Un syst`eme sera consid´r´e fortement coupl´e, si les sous-syst`emes 1 et
2 s’influencent mutuellement de fa¸con significative, par exemple la
thermo-m´ecanique, la thermo-visco´elasticit´e, l’a´ero´elasticit´e, etc. Un syst`eme est
faiblement coupl´e si le sous-syst`eme 1 a une influence significative sur le
sous-syst`eme 2, alors que le sous-syst`eme 2 a une influence mod´er´ee (ou
petite) sur le sous-syst`eme 1, par exemple en a´ero-acoustique. Dans ce cas,
nous pouvons appliquer un sch´ema de r´esolution s´equentiel o nous pouvons
r´esoudre le premier sous-syst`eme 1 (sous-syst`eme 1 en entr´ee), puis nous
r´esolvons le sous-syst`eme 2.
Il existe de nombreuses m´ethodes propos´ees dans la litt´erature afin de
r´esoudre des probl`emes coupl´es. Deux d’entre elles sont les plus couramment
utilis´ees. Le premier est un sch´ema monolithique. Il consiste `a r´esoudre
simultan´ement les ´equations des diff´erents champs simultan´ement (avec un
vite ´enorme, de sorte qu’il pr´esente un cot de calcul ´elev´e. Le second est le
sch´ema ´etag´e. L’objectif d’un sch´ema d’´etag´e est de diviser le probl`eme coupl´e
en un ensemble de sous-probl`emes, donc le sch´ema ´etag´e r´esout diff´erents
champs successivement. Les probl`emes d’une pr´eoccupation majeure pour
nous sont des probl`emes fortement coupl´es o`u les deux ph´enom`enes physiques
s’influencent mutuellement; En particulier, des probl`emes thermom´ecaniques
fortement coupl´es. Dans ce travail, nous nous occupons de l’utilisation de
deux maillages diff´erents pour les parties m´ecanique et thermique afin de
pouvoir tenir compte des ´echelles spatiales propres `a ces deux physiques, nous
consid´erons donc uniquement l’approche ´etag´ee qui permet de s´eparer un
Chapter 2
Strongly coupled problems
2.1
Introduction
The purpose of this chapter is to introduce the problems of interest. First, an introduction to multiphysics is presented in general sense. This allows to introduce different solution schemes for strongly and weakly coupled problems. Then, specific thermo-mechanical problems are tackled starting from the presentation of balance equations. As an example of a constitutive model, thermo-elasticity is presented. Finally, finite element discretization for thermo-mechanical problems is recalled and solution schemes introduced in general setting are applied to thermo-elasticity problem.
2.2
Problems in multiphysics
A problem that involves many physical phenomena influencing each other is called a coupled multiphysics problem. Let’s take for instance the example of a shear band in which mechanical dissipation causes temperature rise which leads the material to soften and then to increase shear strains. To deal with this issue, engineers must combine physical models and algorithms allowing to capture the occuring physical phenomena to simulate the observed effects. For many engineering problems, physical effects of interest result from combination of many physics. Essentially, every problem should be assumed to be coupled unless proven otherwise.
In coupled systems, subsystems interact through interfaces in its general sense, the interaction is called ”one way” if there is no feedback between subsystems. The interaction is called ”two-way” or ”multiway” if there is feedback between sub-systems. We are interested in this latter case, where the response has to be obtained by solving simultaneously the coupled equations which model the system. In computational multi-physics, two types of coupling exist, named strong and weak couplings. Each of those two couplings is linked directly to the tightness of coupling between equations describing the system.
2.2.1
Strong coupling
A system is called strongly coupled, if both sub-system 1 and sub-system 2 have influences on each others, for example thermo-mechanics,
thermo-visco-elasticity, aero-thermo-visco-elasticity, etc. Consider for instance a system, where u1 and
u2 are the two fields. We can write the coupled system u1(t) and u2(t) as:
du1
dt =L1(u1, u2)
du2
dt =L2(u1, u2)
(2.1)
so that functions L1 and L2 are functions of both u1 and u2. In this case we
can apply different kind of schemes to solve the problem :
• Concurrent solution scheme such as monolithic schemes, where we solve simultaneously all equations in one algorithm.
• Staggered schemes, where the coupled problem is split and each field is treated by a different strategy.
• Alternated schemes / Gauss-Seidel approaches for which a fixed point loop is added at each time step on the overall system. This approach should lead to same results as a monolithic scheme, when convergence is obtained.
2.2.2
Weak coupling
A system is weakly coupled if system 1 has significant influence on sub-system 2, while subsub-system 2 has moderate (or small) influence on subsub-system 1, for example in aero-acoustics. In this case, we can apply a sequential solution scheme where we can solve sub-system 1 first (subsystem 1 as an input) and then solve sub-system 2.
L1(u1, u2) ≈ ˆL1(u1) (2.2)
Therefore, equations (2.1) can be rewritten as:
du1 dt ≈ ˆL1(u1) ⇒ u1(t) du2 dt =L2(u1(t), u2) (2.3)
2.3
Solution schemes
2.3.1
Monolithic approach
Consider the interaction between two scalar fields u1 and u2 , where each field
has only one state variable u1(t) and u2(t). The monolithic scheme consists of
resolving simultaneously the fields equations u(t) (in u1 and u2) in one step
(whether implicit or explicit), where u(t) is defined as:
{u(t)} = {u1(t), u2(t)}T (2.4)
{u(0)} = {u0} (2.5)
d{u}
dt =L({u}) (2.6)
Therefore, we obtain a complex system with a very large size, but have the benefit to be unconditional stable for implicit algorithms, meaning that we can go for larger time steps without affecting the stability of the system. On the other hand, explicit algorithms lead to conditional stability, which means
that the stability of the system is influenced by the size of time step, the time
step shall be less than a critical time step ∆t < ∆tcrit. If we denote ∆tcrit1the
critical time step for the stability of the sub-system 1, and ∆tcrit2 the critical
time step for the stability of the sub-system 2, min(∆tcrit1, ∆tcrit2) may be
<< max(∆tcrit1, ∆tcrit2). Moreover, for a monolithic system, the critical time
step for the whole system ∆tcritcan be even smaller than the smallest critical
time steps of each subsystem, that is ∆tcrit << min(∆tcrit1, ∆tcrit2). The
system obtained is very large, and the tangent matrix has the following form:
∂L1 ∂u1 ∂L1 ∂u2 ∂L2 ∂u1 ∂L2 ∂u2 (2.7) The system obtained is generally non-symmetric due to the coupling terms
of ∂L1
∂u2 and
∂L2
∂u1, and this leads to high computational cost generated by
the inversion of the tangent matrix, especially when the coupling is strong, where the sub-diagonal matrices should be taken into account. In fact, when coupling is weak, these matrices can be neglected. Strong coupling may influence convergence, but not inversion time with direct solvers.
2.3.2
Staggered approach
The goal of a staggered scheme is to split the coupled problem into a set of sub-problems, therefore the staggered scheme (also called partitioned (without
a fixed point)) solves different fields separately L({u}) = L2({u}) +L1({u}).
The system becomes simpler due to the reduction of degrees of freedom of each sub-system. Some of the advantages is that we can use the best algorithm of resolution for each sub-system, the best discretization for each sub-system and may use the best time increments for each sub-system [25] (which is not so obvious in practice). One of the disadvantages is that a staggered scheme is not always stable [3]. In case of thermo-mechanical problems, staggered approach can be accomplished in two ways. First, the isothermal staggered scheme in which mechanical part is solved considering isothermal condition for thermal part. Second, the adiabatic staggered scheme in which mechanical part is solved considering adiabatic condition for the thermal part. These approached are examined in detail in the following section.
2.4
Introduction to thermo-mechanics
2.4.1
Balance equations
Let us begin by recalling the continuum balance equations. In the following, u is the displacement field, T is the external temperature, Θ is the internal temperature and F is the deformation gradient tensor: grad(u).
Principle of mass conservation states that mass can neither be created nor be destroyed. Therefore, mass of a material point dm always remains the same.
dm = ρ0dV0 = ρdV (2.8)
where ρ0 is the initial material density, dV0 is the initial volume, ρ is the
material density and dV is the volume of the material point after deformation. Equation (2.8) can be rewritten in terms of the deformation gradient tensor F as follows:
ρ det [F] = ρ0 (2.9)
where, det [F] represents the change in volume of the material point.
Principle of conservation of linear momentum is obtained from Newton’s second law of motion which states that sum of the forces acting on a body is equal to the rate of change of momentum of the body, which yeilds in local form:
ρ ˙v = ρb + div{σ} (2.10)
where v is the velocity of the particle, b are the applied body forces and σ is the Cauchy stress tensor, the divergence of which gives internal forces.
Conservation of angular momentum imposes that Cauchy stress tensor σ should be symmetric. That is,
σ = σT (2.11)
or equivalently,
PFT = FPT (2.12)
The first law of thermodynamics states that energy can be transformed, but cannot be created nor destroyed. It is usually formulated as a balance of internal energy by stating that the change in the internal energy of a system is equal to the amount of heat supplied to the system, minus the amount of work performed by the system on its surroundings, in other words it is a transformation from heat energy to mechanical energy and vice-versa. It can be written as:
ρ ˙U = P : ˙F + ρ0r − div0{q} (2.13)
where U is the internal energy density (per unit volume), q is the heat flux,
and div0 is divergence with respect to reference co-ordinates.
The second law of thermodynamics is rather an evolution principle than a balance law. It is an expression of the tendency that over time, differences in temperature, pressure, and chemical potential equilibrate in an isolated physical system. From the state of thermodynamic equilibrium, the law deduced the principle of the increase of entropy and explains the phenomenon of irreversibility in nature. The first law of thermodynamics provides the basic definition of thermodynamic energy (or internal energy), associated with all thermodynamic systems, and states the rule of conservation of energy in nature. However, the concept of energy in the first law does not account for the observation that natural processes have a preferred direction of progress. For example, spontaneously, heat always flows to regions of lower temperature, never to regions of higher temperature without external work being performed on the system. The first law is completely symmetrical with respect to the initial and final states of an evolving system. The key concept for the explanation of this phenomenon through the second law of thermodynamics is the definition of a new physical property, the entropy S, defined as :
S = Z
Ω
ρηdΩ (2.14)
where η is the specific entropy. The second law of thermodynamics can be represented as follows: ˙ S − Z Ω ρr T dΩ − Z ∂Ω q · n T dS ≥ 0 (2.15)
which can be rewritten as: Z Ω n ρ ˙η − ρr T + div q T o dΩ ≥ 0 (2.16)
As the domain Ω is arbitrary, one can obtain Clausius-Duhem inequality from equation (2.16) and (2.13) as follows:
˙ΓT = ρT ˙η + σ : D − ρ ˙U − 1
Tq · grad{T } ≥ 0 (2.17)
where ˙ΓT represents the dissipated power.
2.4.2
Thermo-elasticity example
As an example of a strongly coupled thermo-mechanical problem, let us consider thermo-elasticity. In this case, the stress tensor σ is related to both,
the elastic strain and temperature rise θ = T − Tref as follows:
σ = E : ( − αθ) (2.18)
where E is the elasticity tensor, Tref a reference temperature, and α is
the coefficient of thermal expansion tensor. Rise in temperature causes the material to expand without inducing any stress. Therefore, strain responsible for stress in the body is the difference between the total strain and strain caused due to the thermal expansion of the material. This very phenomenon is explained by the equation (2.18). Now, one can write the linear momentum conservation equation (2.10) for thermo-elasticity as follows:
ρ ˙v = div {E : ( − αθ)} + ρb (2.19)
Equation (2.19) represents the effect of thermal part on the mechanical part for thermo-elasticity. On the other hand, expansion of material causes its temperature to decrease and contraction causes its temperature to rise. One can write the thermal problem as follows in order to incorporate this effect:
˜
c ˙θ = div{ ˜K · grad(θ)} − α : E : ˙ + r
Tref
(2.20)
where r is the external heat source, c is the heat capacity and ˜c = c
Tref,
similarly, K is the thermal conductivity tensor and ˜K = TK
(2.19) and (2.20) represent the strong form of coupled thermo-mechanical
problem.1 One can obtain the weak form easily by multiplying the equations
with test functions and integrating over the domain. For time integration, one can consider Newmark scheme:
an+1= un+1− un β∆t2 − vn β∆t− 0.5 − β β an vn+1 = vn+ (1 − γ)∆tan+ γ∆tan+1 (2.21)
here subscripts represent time step considered, a = ˙v = ¨u the acceleration, β
and γ are the algorithm parameters. For good stability, integration parameters
are chosen as β = 1
4 and γ =
1
2. For finite element space discretization, we
consider finite element interpolation:
u = N nodes X i=1 Ni(x)ui θ = N nodes X i=1 Ni(x)θi (2.22)
where Ni(x) are the nodal shape functions, ui are nodal displacements and
θi are nodal temperatures. Mass matrix can be given from finite element
interpolation as a function of shape function matrix [Ne] as follows:
[Me] = nElements X i=1 Z Ωe ρ[Ne]T[Ne]dΩ (2.23)
The stiffness matrix is given as derivative of shape function matrix with
respect to spatial coordinates [De]:
[E] = nElements X i=1 Z Ωe [De]T[E][De]dΩ (2.24)
In case of thermo-elasticity problem we have the coupling matrix given as follows: [B] = nElements X i=1 − Z Ωe [De]T[αE][Ne]dΩ (2.25)
Capacity matrix is similar to mass matrix and it is given by:
[ ˜C] = nElements X i=1 Z Ωe c Tref [Ne]T[Ne]dΩ (2.26)
Conductivity matrix is similar to stiffness matrix and is given by:
[ ˜K] = nElements X i=1 Z Ωe 1 Tref [De]T[K][De]dΩ (2.27)
Using the matrices defined above, the mechanical part of the finite element problem can be written as follows:
1 β∆t2[M ]{un+1} + [E]{un+1} + [B]{θn+1} = 1 β∆t2[M ]{un} + 1 β∆t[M ]{{vn} + (0.5 − β)∆t{an}} + {b} + {t} (2.28)
Whereas, the thermal problem is given as:
[B]T{un+1}−[ ˜C]{θn+1}−∆t[ ˜K]{θn+1} = −[ ˜C]{θn}+[B]T{un}− r Tref + ∆t Tref {qn} (2.29) 2.4.2.1 Monolithic approach
In the monolithic approach, we solve both the mechanical and thermal problems simultaneously. The problem can be stated as follows:
1 β∆t2[M ] + [E] [B] [B]T −[ ˜C] − ∆t[ ˜K] {un+1} {θn+1} ={Fu} {Fθ} (2.30)
Where, {Fu} and {Fθ} can be given as follows: {Fu} = [M ] β∆t2{un} + [M ] β∆t{{vn} + (0.5 − β)∆t{an}} + {b} + {t} {Fθ} = −[ ˜C]{θn} + [B]T{un} − {r} Tref + ∆t Tref {qn} (2.31) Velocity and acceleration can be given as represented by equation 2.21. As explained earlier, this approach is unconditionally stable. However, if one uses monolithic approach, the problem size is very big, therefore more com-putational power is needed. Another disadvantage is that one can not use two different meshes for mechanical and thermal part while using monolithic approach. This is mainly because of the inability to construct a single coupling matrix [B] on two different meshes.
2.4.2.2 Staggered approach with isothermal split
In a staggered approach, thermal and mechanical problems are split and then are solved one by one. The simplest staggered technique is to consider isothermal split in which first the mechanical problem is solved assuming no variation in temperature and then the thermal problem is solved at fixed geometry: 1 β∆t2[M ] + [E] {un+1} = {Fu} − [B]{θn} (2.32) h −[ ˜C] − ∆t[ ˜K]i{θn+1} = {Fθ} − [B]T{un+1} (2.33)
The advantage of using isothermal split is that it is very simple to implement, one can use different meshes for mechanical and thermal part. This allows us to divide one big problem into two smaller problems. The main limitation of using isothermal split is that the algorithm is not unconditionally stable as shown by Simo et al.[3].
2.4.2.3 Staggered approach with adiabatic split
In order to have a stable algorithm, one can use an adiabatic split in which, while solving the mechanical part, adiabatic thermal conditions are assumed. It is represented as follows: 1 β∆t2[M ] + [E] [B] [B]T −[ ˜C]] {un+1} {θad} ={Fu} {Fθ} (2.34) h −[ ˜C] − ∆t[ ˜K]i{θn+1} = {Fθ} − [B]T{un+1} (2.35)
Note that in the mechanical step, one does not need to calculate θad since it
is not needed in the calculations. This split makes the algorithm stable when used with a constant time step [3].
When using a staggered scheme with different meshes for mechanical and thermal part, first thermal fields need to be interpolated on mechanical mesh and then mechanical problem is solved. In this step, the coupling matrix [B] is constructed only on the mechanical mesh. Then, the calculated
displacement field {un+1} is interpolated on the thermal mesh and then the
thermal problem is solved. Here, the coupling matrix [B] is constructed only on the thermal mesh. Note that we only perform interpolations from one mesh to the other, no extrapolations are performed.
2.5
Conclusion
This chapter introduced problems in multiphysics and relevant solution schemes in general. Weak coupling and strong coupling in multiphysics was presented. The problems of prime concern for us are strongly coupled problems where both physical phenomena influence each other.
Particularly, we focus on thermo-mechanical problems which were pre-sented in the final section, with example of thermo-elasticity, monolithic and staggered approaches were presented. In this work, we are concerned with using two different meshes for mechanical and thermal parts in order to have separate mesh adaptation, therefore we only consider the staggered approach which allows separation of a coupled problem in subproblems related to each physical phenomena.
The proposed mesh adaptation is based on variational principle which is introduced in the following chapter.
![Figure 4.5: Original algorithm of Rivara [61] based on target triangle. Target triangle is t 0 .](https://thumb-eu.123doks.com/thumbv2/123doknet/7901779.264601/70.892.165.776.189.386/figure-original-algorithm-rivara-target-triangle-target-triangle.webp)
![Figure 5.29: Geometry of the test case [31].](https://thumb-eu.123doks.com/thumbv2/123doknet/7901779.264601/99.892.274.609.192.534/figure-geometry-of-the-test-case.webp)
